Abstract
We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that, in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. V. Dolgachev and V. A. Iskovskikh, “Finite subgroups of the plane Cremona group,” in Algebra, Arithmetic, and Geometry, Progr. Math. (Boston, MA, Birkhäuser Boston, 2009), Vol. 269, pp. 443–548.
A. Beauville, “\(p\)-elementary subgroups of the Cremona group,” J. Algebra 314 (2), 553–564 (2007).
Yu. Prokhorov, “2-elementary subgroups of the space Cremona group,” in Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat. (Springer, Cham, 2014), Vol. 79, pp. 215–229.
Yu. Prokhorov, “\(p\)-elementary subgroups of the Cremona group of rank 3,” in Classification of Algebraic Varieties, EMS Ser. Congr. Rep. (Eur. Math. Soc., Zurich, 2011), pp. 327–338.
Yu. Prokhorov and C. Shramov, “\(p\)-subgroups in the space Cremona group,” Math. Nachr. 291 (8-9), 1374–1389 (2018).
Yu. Prokhorov, “\(G\)-Fano threefolds. I,” Adv. Geom. 13 (3), 389–418 (2013).
Y. Prokhorov, “On the number of singular points of terminal factorial Fano threefolds,” Math. Notes 101 (6), 1068–1073 (2017).
A. G. Kuznetsov, Yu. G. Prokhorov, and C. A. Shramov, “Hilbert schemes of lines and conics and automorphism groups of Fano threefolds,” Jpn. J. Math. 13 (1), 109–185 (2018).
Vl. Popov, “Jordan groups and automorphism groups of algebraic varieties,” in Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat. (Springer, Cham, 2014), Vol. 79, pp. 185–213.
Yu. Prokhorov and C. Shramov, “Jordan property for groups of birational selfmaps,” Compos. Math. 150 (12), 2054–2072 (2014).
W.-L. Chow, “On the geometry of algebraic homogeneous spaces,” Ann. of Math. (2) 50 (1), 32–67 (1949).
M. Hall Jr., The Theory of Groups (Chelsea Publ., New York, 1976).
A. Schweizer, “On the exponent of the automorphism group of a compact Riemann surface,” Arch. Math. (Basel) 107 (4), 329–340 (2016).
V. A. Iskovskikh and Y. Prokhorov, “Fano varieties,” in Algebraic Geometry, V, Encyclopaedia Math. Sci. (Springer, Cham, 1999), Vol. 47.
M. Reid, “Young person’s guide to canonical singularities,” in Algebraic Geometry, Bowdoin, 1985, Part 1 Proc. Sympos. Pure Math. (Amer. Math. Soc., Providence, RI, 1987), Vol. 46, pp. 345–414.
Yo. Namikawa, “Smoothing Fano 3-folds,” J. Algebraic Geom. 6 (2), 307–324 (1997).
Y. Kawamata, “Boundedness of \(\mathbb{Q}\)-Fano threefolds,” in Proceedings of the International Conference on Algebra, Contemp. Math., Part 3 (Amer. Math. Soc., Providence, RI, 1992), Vol. 131, pp. 439–445.
Yu. Prokhorov, “\(G\)-Fano threefolds. II,” Adv. Geom. 13 (3), 419–434 (2013).
K. Shin, “\(3\)-dimensional Fano varieties with canonical singularities,” Tokyo J. Math. 12 (2), 375–385 (1989).
Sh. Mukai, “Curves, K3 surfaces and Fano 3-folds of genus \(\le 10\),” in Algebraic Geometry and Commutative Algebra (Kinokuniya, Tokyo, 1988), Vol. I, pp. 357–377.
Yu. Prokhorov, Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I, arXiv: 1907.05678 (2019).
V. Iskovskikh, “Fano 3-folds. I,” Math. USSR-Izv. 11 (3), 485–527 (1977); “Fano 3-folds. II ,” Math. USSR-Izv. 12 (3), 469–506 (1978).
O. Debarre and A. G. Kuznetsov, “Gushel–Mukai varieties: classification and birationalities,” Algebr. Geom. 5 (1), 15–76 (2018).
W. Feit, “The current situation in the theory of finite simple groups,” in Actes du Congrès International des Mathématiciens, Tome 1 (Gauthier-Villars, Paris, 1971), pp. 55–93.
A. Borel, “Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes,” Tohoku Math. J. (2) 13, 216–240 (1961).
A. Borel and J.-P. Serre, “Sur certains sous-groupes des groupes de Lie compacts,” Comment. Math. Helv. 27, 128–139 (1953).
K. Tahara, “On the finite subgroups of \(\operatorname{GL}(3,\mathbb{Z})\),” Nagoya Math. J. 41, 169–209 (1971).
Acknowledgments
The author wishes to express gratitude to her advisor C. Shramov for suggesting this problem, as well as for patience and invaluable support. Thanks are also due to A. Avilov, Yu. Prokhorov, and A. Trepalin for useful discussions.
Funding
This work was supported by the Russian Science Foundation under grant 18-11-00121.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kuznetsova, A.A. Finite 3-Subgroups in the Cremona Group of Rank 3. Math Notes 108, 697–715 (2020). https://doi.org/10.1134/S0001434620110085
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620110085